A Holographic Perspective on Supersymmetry in Curved Space Alberto Zaffaroni Universit` a di Milano-Bicocca SUPERFIELDS, 2012, Pisa Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved Space SUPERFIELDS, 2012, Pisa 1 / 29
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A Holographic Perspective on Supersymmetry inCurved Space
Alberto Zaffaroni
Universita di Milano-Bicocca
SUPERFIELDS, 2012, Pisa
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 1 / 29
Outline
Outline
I Introduction and Motivations
I Holographic Perspective
I SCFT on Curved Backgrounds
I Conclusions and Open Problems
based onC. Klare, A. Tomasiello, A. Z. arXiv:1205.1062D.Cassani, C. Klare, D. Martelli, A. Tomasiello, A. Z. arXiv:1207.2181
some related results inT. Dumitrescu, G. Festuccia, N. Seiberg arXiv:1205.1115
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 2 / 29
Outline
Outline
I Introduction and Motivations
I Holographic Perspective
I SCFT on Curved Backgrounds
I Conclusions and Open Problems
based onC. Klare, A. Tomasiello, A. Z. arXiv:1205.1062D.Cassani, C. Klare, D. Martelli, A. Tomasiello, A. Z. arXiv:1207.2181
some related results inT. Dumitrescu, G. Festuccia, N. Seiberg arXiv:1205.1115
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 2 / 29
Outline
Outline
I Introduction and Motivations
I Holographic Perspective
I SCFT on Curved Backgrounds
I Conclusions and Open Problems
based onC. Klare, A. Tomasiello, A. Z. arXiv:1205.1062D.Cassani, C. Klare, D. Martelli, A. Tomasiello, A. Z. arXiv:1207.2181
some related results inT. Dumitrescu, G. Festuccia, N. Seiberg arXiv:1205.1115
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 2 / 29
Introduction and Motivations
Recent interest in the very basic question:
When and How we can Define Superconformal Field Theories and RigidSupersymmetric Field Theories on Nontrivial Spacetimes
I Familiar examples: Rp,q, AdSd , Rp,q × T s ,... More general analysisjust started ...
Festuccia, SeibergSantleben, TsimpisKlare, Tomasiello, A. Z.Dumitrescu, Festuccia, SeibergCassani, Klare, Martelli, Tomasiello, A. Z.Liu, Pando Zayas, Reichmann[...]
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 3 / 29
Introduction and Motivations
We recently learned more on how to put supersymmetric theories on curvedmanifolds and how to compute exact quantities.
I the partition function of 3d N = 2 theory on a three sphere
(F ∼ N3/2, F-theorem, maximization at exact R-symmetry)
I many other computation on round and squashed spheres in 3,4,5 d(applications to the elusive (2,0), AGT,...)
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 4 / 29
Introduction and Motivations
We also found new examples of regular asymptotically AdS backgrounds
ds2d+1 =
dr2
r2+ (r2ds2
Md+ O(r))
which describe SCFTs on the curved space Md
I AdSd+1 boundary is Rd,1 or Sd × R, depending on r-foliation; differentcoordinates in bulk may just correspond to Weyl rescaling on boundary.
I Other backgrounds with non conformally flat, non Einstein boundary →CFTs on curved space-times
I Examples of 3d SCFTs on squashed spheres: N3/2 free energies[Martelli,Passias,Sparks]
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 5 / 29
Holographic Perspective
CFTs on Curved spaces
General vacua of a bulk effective action
L = −1
2R+ FµνF
µν + V ...
with a metric
ds2d+1 =
dr2
r2+ (r2ds2
Md+ O(r)) A = AMd
+ O(1/r)
and a gauge fields profile, correspond to CFTs on a d-manifold Md and anon trivial background field for the R-symmetry
LCFT + JµAµ
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 6 / 29
Holographic Perspective
SCFTs on Curved spaces
Requiring that some supersymmetry is preserved: [...,klare,tomasiello,A.Z.](∇A
M +1
2γM +
i
2�FγM
)ε = 0 ∇A
M ≡ ∇M − iAM
Near the boundary:
ε = r12 ε+ + r−
12 ε−
(∇a − iAa)ε+ + γaε− = 0 =⇒ ∇Aa ε+ =
1
dγa�∇Aε+
Existence of a Conformal Killing Spinor.
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 7 / 29
Holographic Perspective
SCFTs on Curved spaces
We can understand this by coupling the CFT to background fields ofconformal supergravity gmn, ψm and Am:
−1
2gmnT
mn + AmJm + ψmJm
In order to preserve some supersymmetry, the gravitino variation mustvanish.
δψm = (∇m − iAm) ε+ + γmε− = 0
ε± parameters for the supersymmetries and the superconformaltransformations.
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 8 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Curved spaces
The condition for preserving some supersymmetry is then
∇Aa ε+ =
1
dγa�∇Aε+
I (twisted) conformal Killing equation
I projection of ∇Aa on the irreducible spin 3/2 component
I conformally invariant equation
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 9 / 29
Supersymmetry on Curved Backgrounds
Twistors
Conformal Killing Spinors
∇aε =1
4γa�∇ε
with A = 0 (also called twistors) have been studied and classified:
I Lorentzian: pp-waves and Fefferman metrics.
I Euclidean: conformally equivalent to manifolds with Killing spinors
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 10 / 29
Supersymmetry on Curved Backgrounds
Euclidean Twistors
In the Euclidean a Conformal Killing Spinor with A = 0 becomes a KillingSpinor on a Weyl rescaled metric [Lichnerowitz]:
∇aε =1
dγa�∇ε =⇒ ∇aε = µγaε
Manifolds with Killing Spinors are in turn classified: in the compact case thecone over it has restricted holonomy:
dim H C (H)
3 S3 R4
4 S4 R5
5 Sasaki-Einstein CY3
6 Nearly-kahler G2 manifolds
or quotients...
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 11 / 29
Supersymmetry on Curved Backgrounds
The result for A 6= 0Focus on a single solution of the CKS eqs in 4d: ∇A
Euclidean j , ω dω = W ∧ ω[klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg]
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 12 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Lorentzian Curved Backgrounds
Information on a 4d spinor ε+ of given chirality can be encoded in differentialforms (bilinears). In Lorentzian signature from a CKS ε+ we can construct a realnull vector z and a complex two form ω:
ε+ ⊗ ε+ = z + i ∗ z , ε+ ⊗ ε− ≡ ω = z ∧ w , z2 = 0
z = e0 − e1 , w = e2 + ie3
The CKS equation translates into a set of linear constraints for the differential ofthe bilinears and the gauge field
∇Aa ε+ =
1
dγa�∇Aε+ =⇒ linear eqs for {dz , dω,A}
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 13 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Lorentzian Curved Backgrounds
The existence of a charged CKS is equivalent to the existence of a nullCKV.
∇Aa ε+ =
1
4γa�∇Aε+ =⇒ ∇µzν +∇νzµ = λgµν (8 conditions )
(12 conditions) real gauge field A (4 conditions)
Whenever there exists a null CKV a SCFT preserves some supersymmetryin curved space. [cassani,klare,martelli, tomasiello,A.Z.]
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 14 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Lorentzian Curved Backgrounds
Every conformal Killing vector becomes Killing in a Weyl rescaled metric
Lzgµν = λgµν ⇒ Lz(e2f gµν) = (λ+ 2z · df )gµν
We can then choose adapted coordinates z = ∂/∂y
ds2 = 2H−1(du + βmdxm)(dy + ρmdx
m + F (du + βmdxm)) + Hhmndx
mdxn
where H, hmn, βm, ρm do not depend on y. A is completely determined by thesefunctions.
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 15 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Euclidean Curved Backgrounds
Information on a 4d spinor ε+ of given chirality can be encoded in differentialforms (bilinears). In Euclidean signature from a CKS ε+ we can construct twotwo forms j , ω:
ε+ ⊗ ε†+ =1
4eBe−i j , ε+ ⊗ ε+ =
1
4eBω , eB ≡ ||ε+||2
j = e1e2 + e3e4 , ω = (e1 + ie2)(e3 + ie4)
The CKS equation translates into a set of linear constraints for the differential ofthe bilinears and the gauge field
∇Aa ε+ =
1
dγa�∇Aε+ =⇒ linear eqs for {dj , dω,A}
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 16 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Euclidean Curved Backgrounds
The existence of a CKS is (locally) equivalent to the existence of a complexstructure.
CKS =⇒w3 = 0 (complexmanifold)
A1,0 = −i(−1
2w5
0,1 +1
4w4
1,0 +1
2∂B)
A0,1 = −i(+1
2w5
0,1 −3
4w4
0,1 +1
2∂B)
I dj = w4 ∧ j , dω = w5 ∧ ω + w3 ∧ ωI Notice that A is in general complex
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 17 / 29
Supersymmetry on Curved Backgrounds
SCFTs on Euclidean Curved Backgrounds
Examples of manifolds supporting supersymmetry
I Kahler manifolds
I Complex but not Kahler (S3 × S1 , superconformal index, A = idφ)
I Subtelties: S4 is not complex. Decomposing the S4 Killing spinorε = ε+ + ε−: ε+ vanish at North pole: R4 = S4 −NP is complex.
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 18 / 29
Supersymmetry on Curved Backgrounds
Supersymmetry on Curved spaces
More generally, we may ask when we can put a generic supersymmetrictheory on a curved background: coupled it the Poincare supergravity andset the gravitino variation to zero. [festuccia,seiberg]
A theory with an R-symmetry can be coupled to new minimal supergravitywhich has two auxiliary fields (aµ, vµ) with d(∗v) = 0. The gravitinovariation is:
∇mε+ = −i(
1
2vnγnm + (v − a)m
)ε+
Alberto Zaffaroni (Milano-Bicocca) A Holographic Perspective on Supersymmetry in Curved SpaceSUPERFIELDS, 2012, Pisa 19 / 29
Supersymmetry on Curved Backgrounds
Supersymmetry on Curved spaces
Conformal Killing spinors are closely related to solutions of the new minimalcondition. Defining �∇Aε+ ≡ 2i�vε+ we can map a CKS to (and viceversa)
∇Aa ε+ −
1
4γa�∇Aε+ = 0 =⇒ ∇mε+ = −i
(1
2vnγnm + (v − a)m
)ε+
Condition for coupling a supersymmetric theory to new minimal supergravity(gµν , aµ, vµ) same as the condition for existence of a CKS with a ≡ A + 3
2v
I One Euclidean supercharge ε+: M4 should be a complex manifold